The study of local cohomology groups has its origin in the observation, already implicit in serre s paper fac, that many statements about projective varieties can be reformulated in terms of graded rings, or complete local r this a 1 lows one to conjecture and then prove sta about local rings, which then. Motivic cohomology is usually thought of as the universal blochogus cohomology, while the universal weil cohomology should probably be pure motives with respect to rational equivalence probably, because it depends on what exactly you mean by universal and weil cohomology. The purpose of this paper is to interpret rigid syntomic cohomology, defined by amnon besser bes, as a padic absolute hodge cohomology. We shall call an abelian group aa gmodule if it is a left zgmodule. To accomplish these tasks, we define and study the proetale topology, which is a. So, one way to think about homology and cohomology is that they are ways of counting the numb.
Rationality and the functional equation via weil cohomology29 4. On adic compatibility for abelian motives the mumford tate. The morphism f acts on the ith cohomology space hivofv as a linear map denoted by fforh one can take a p adic cohomology, e. Bounded cohomology of lattices in higher rank lie groups received july 14, 1998. On the adic cohomology of varieties over number fields. Cohomology plays a fundamental role in modern physics. The cohomology of the sheaf of witt vectors gives the slope cohomology 1, 11. A brief introduction to adic cohomology32 chapter 5. The concrete interpretation of the cochain complex as a discretization of differential forms was a key insight of thom and whitney from the 1950s. The zetafunction of an algebraic variety over a finite field can be expressed in terms of a frobenius operator acting on p adic cohomology groups of this variety. Cohomology groups synonyms, cohomology groups pronunciation, cohomology groups translation, english dictionary definition of cohomology groups. Istrodlction in this paper i shall describe a method of associating a spectrum, and hence a cohomology theory, to a category with a compositionlaw of a. Here is an article by tim gowers, discussing combinatorics and cohomology.
University of groningen on padic monodromy stienstra, jan. Istrodlction in this paper i shall describe a method of associating a spectrum, and hence a cohomology theory, to a category with a compositionlaw of a suitable kind. Cohomology of groups 5 a mapping problem of the following form. Fultons trace formula for coherent sheaf cohomology39 5.
The idea being that if one has information about the open sets that make up a space as. In that cohomology is an important part of the story of higher category theory as revolution, we should expect it to show itself in the development of current physics. N2end kv is a nilpotent endomorphism, such that for all x2w f the condition. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. For open varieties go yamashita this text is a report of a talk padic. If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. The morphism f acts on the ith cohomology space hivofv as a linear map denoted by fforh one can take a padic cohomology, e. The idea to prove these results is to embed the cohomology of bung into the cohomology of the stack of principal gbundles together with. That is, write the homology in this degree as a direct sum of a free module and a torsion module. The padic hodge theory of semistable galois representations. This is a list of some of the ordinary and generalized or extraordinary homology and cohomology theories in algebraic topology that are defined on the categories of cw complexes or spectra.
A new padic cohomology theory in characteristic p work in progress by jeanmarc fontaine and uwe jannsen regensburg, 17 february 2011 conference arithmetic and motivic algebraic geometry uwe jannsen university of regensburg padic cohomology in characteristic p 17 february 2011 1 27. Commutative algebra and the cohomology of groups, iii. Singular homology with coefficients 3 given a topological space x, we denote its singular chain complex by cx 2ch 0z. Zeta functions in algebraic geometry mircea mustat. In these expository notes derived categories and functors are gently introduced, and used along with koszul complexes to develop the basics of local cohomology. Syntomic cohomology as a padic absolute hodge cohomology. We study in particular the case d 2, and compute the euler characteristic. There is a classical relation between the padic absolute value of the eigenvalues of frobenius on crystalline cohomology and hodge numbers, for a variety in characteristic p. Weil cohomology theories and the weil conjectures23 4. Bounded cohomology of lattices in higher rank lie groups. Rigidity and frobenius structure richard crew the university of florida march 16, 2015 introduction the purpose of this note is show that an irreducible rigid di erential equation on an open subset of p1 with regular singularities and rational exponents has, with reasonable local assumptions relative a prime p, a frobenius structure relative. For each prime number we may form the adic etale cohomology group hi x hi et xk. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and cech cohomology 15.
Nevertheless, the cohomology of a space, which is obtained by dualizing its simplicail chain complex, carries important additional structure. Commelin december 11, 2014 1 prolegomena let gbe a group. We refer to this by saying that cx is levelwise free. B 0 sends g to g 1, and such things generate as a left ideal the kernel of b 0. Let a be an abelian category, that is, roughly, an additive category in which there exist wellbehaved kernels and cokernels for each morphism, so that, for example, the. This item does not appear to have any files that can be experienced on. Cohomology theories, and more specifically algebraic structures on the cochain complex, have recently surfaced in unexpected areas of applied mathematics. Cohomology is a very powerful topological tool, but its level of abstraction can scare away interested students. M then the equivialence class that contains is denoted and called the cohomology class of 3. Recall that cx consists of the singular chain groups c kx which are free abelian groups in each degree.
Rigid cohomology does seem to be a universal padic cohomology with. X y determines a homomorphism from the cohomology ring of y to that of x. On the adic cohomology of varieties over number fields and its galois cohomology uvve jannsen if x is a smooth, projective variety over a number field fc, then the absolute galois group gk galfcfc acts on the etale cohomology groups hlx, qi1en, where x x xf c jc for an algebraic closure jc of k. Thenumberoff qrational points on v is related to the action of f. Ill go about as intuitive and nobackgroundassumed as i can come up with, which likely is still too technical for most and too naive for most others. A key theme is the explicit, computable nature of these constructions. An introduction to the cohomology of groups peter j. Bbounded cohomology associated to an extension of groups equipped with length functions, and the serre spectral sequence in bbounded cohomology for developable complexes of groups. As with homology we start with the algebraic set up.
No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology. In what follows, cohomology is taken with coefficients in the integers z, unless stated otherwise. Topological hochschild homology and integral padic hodge theory bhargav bhatt, matthew morrow and peter scholze abstract. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Lectures on local cohomology university of illinois at. Tomoyuki abe kavli institute for the physics and mathematics of the universe. From the long exact cohomology sequence associated to the lower short exact sequence we obtain an element. Cohomology groups definition of cohomology groups by the. When k is of characteristic p, then the natural replacement for padic galois representations that one encounters when looking at the cohomology of varieties over k are phi,nabla modules over the amice ring, and the theory that produces them is rigid cohomology.
The solid arrows represent given maps, with the composite f m m00 equal to the zero map, and the dotted arrow represents a map we want to construct. When z, q, qz are considered as gmodules, this is with the. University of groningen a generalized artihmetic geometric. Uvadare digital academic repository the cohomology rings. This is a padic analogue of a work of beilinson be1 which interprets beilinsondeligne cohomology in terms of absolute hodge cohomology. For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3. Base change in this section we show that the ladic variants of our local systems commute with base change and we conclude that the euler characteristic e cm 3. This is a long theory to do everything, but one computation of ext r goes as follows. The jacquetlanglands correspondence realized on the cohomology 42 4.
But nowadays there is a whole scala of cohomologies that are weil cohomologies or at least share enough properties with weil cohomologies to warrant further study. Tate, galois cohomology we say g acts trivially on a if. As far as i know, the aim was to prove the weil conjectures, which deligne did in 1974. Alternatively, is a module over the ring in the usual sense of module. Lectures on local cohomology and duality joseph lipman abstract.
Our aim is to provide an overview of the main constructions and some useful techniques of proof, not to give a complete account of the theory. For example, with the trivial action is a module over any group, as is for any positive. Let fbe a p adic eld, with residue eld of cardinality q. Let rep k g denote the category of representations of gover k. Local duality and its farreaching generalization, greenleesmay duality, are treated. Completed cohomology and the adic lang lands program. As a second year graduate textbook, cohomology of groups introduces students to cohomology theory involving a rich interplay between algebra and topology with a minimum of prerequisites. Unlike more subtle invariants such as homotopy groups, the. Modern applications of homology and cohomology institute. The first four sections of this paper give a survey of the papers of monsky and. Galois representations can be constructed by taking the. Pdf cohomology of siegel varieties with padic integral. These choices determine an embedding of b st, the ring of periods of semistable varieties, in b dr.
N, where v is a nitedimensional smooth representation of w f over k. Fundamental physics is all controled by cohomology. Given a group gthere exists a con nected cw complex xwhich is aspherical with. In the process, we will define a theory of padic absolute hodge cohomology with. R when we refer to cohomology, even though it may be coming from forms. Lectures on ncategories and cohomology talks by john baez, notes by michael shulman contents preface 2 1.
This means that is an abelian group equipped with a left action of, i. It requires no prior knowledge of the concepts of algebraic topology or cohomology. The cohomology ring of a point is the ring z in degree 0. Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. Lectures on local cohomology craig huneke and appendix 1 by amelia taylor abstract. As an application, i will show that the dmodule theoretic pushforward of an isocrystal along a proper smooth morphism is an isocrystal. As the prime varies, these galois representations have a lot of. Cohomology of varieties with totally degenerate reduction 265 we denote by b dr the ring of padic periods of fontaine. As has been noted by meyer in m2, the category in which one does homological algebra in the bornological framework is almost never abelian. For other sorts of homology theories see the links at the end of this article. In this paper i study some properties of these gkmodules. I will be glad if moreover someone can point out a reference about this i have looked at milnes notes and book and lei fus book, but they dont talk about this. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g.
Hochschild cohomology of abelian categories and ringed. Algebraic topology of random fields and complexes researchthesis as partial ful. On adic compatibility for abelian motives the mumford. Some of these appear in chapter 3, where we compare. University of groningen on padic monodromy stienstra. This book offers a selfcontained exposition to this subject and to the theory of characteristic classes from the curvature point of view.